The Skin Effect Section 60

Long thin straight wire with AC current A variable finite current flows inside the wire, j =  E. At high frequency, j is concentrated at the surface in a thin layer. This is the “skin effect.”

Wire with circular cross section By symmetry, E is distributed uniformly around the surface of the wire. The value of E is time dependent. In the low-frequency quasistatic case, ignore the time derivatives of fields outside. Then E just outside conductor is also spatially uniform: E = E 0 (t) e z Since  = 0 E t is continuous.

Instantaneous total current Since we ignore in the quasi-static case outside of the conductor in External magnetic field:

Inside the wire, we don’t ignore the time derivatives of the fields. Cylindrical coordinates Periodic field See appendix

A second order differential equation has two independent solutions. Keep the one that is finite at r = 0. It only needs to hold out to r = a. Same distribution for j =  E. Bessel function of complex argument

Still inside the wire… There is only a  component to the curl. Cylindrical coordinates Since  = 1

Find the constant from the condition H(a) = 2J/ca. It depends on Total current

Low frequencies E(r) increases toward the surface. H = 0 at the center.

High frequencies Keep only the rapidly varying exponential factor valid over most of the cross section large

These equations are the same as (59.3)-(59.5) for the field near the surface of a conductor of any shape when the penetration depth is small. Both E and H are small at the center (r = 0) and maximum on the surface r = a.

What happens to an AC current in a wire as the frequency is increased? 1.The current becomes concentrated at the center. 2.The current becomes concentrated on the surface. 3.The current oscillates between center and surface